5 results
Search Results
Now showing 1 - 5 of 5
- ItemStability of deep neural networks via discrete rough paths(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bayer, Christian; Friz, Peter; Tapia, NikolasUsing rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks. In particular we derive stability bounds in terms of the total p-variation of trained weights for any p ≥ 1.
- ItemPricing under rough volatility(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bayer, Christian; Friz, Peter; Gatheral, JimFrom an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum [SSRN 2509457, 2014] previously showed that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated volatility. We analyze in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.
- ItemUnified signature cumulants and generalized Magnus expansions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Friz, Peter; Hager, Paul; Tapia, NikolasThe signature of a path can be described as its full non-commutative exponential. Following T. Lyons we regard its expectation, the expected signature, as path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants, and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions; with motivation ranging from financial mathematics to statistical physics. From an affine process perspective, the functional relation may be interpreted as infinite-dimensional, non-commutative (``Hausdorff") variation of Riccati's equation. Many examples are given.
- ItemPathwise McKean--Vlasov theory with additive noise(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Coghi, Michele; Deuschel, Jean-Dominique; Friz, Peter; Maurelli, MarioWe take a pathwise approach to classical McKean-Vlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [34]. Our study was prompted by some concrete problems in battery modelling [19], and also by recent progress on rough-pathwise McKean-Vlasov theory, notably Cass--Lyons [9], and then Bailleul, Catellier and Delarue [4]. Such a ``pathwise McKean-Vlasov theory'' can be traced back to Tanaka [36]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4, 9, 36]. As novel applications we discuss mean field convergence without a priori independence and exchangeability assumption; common noise and reflecting boundaries. Last not least, we generalize Dawson--Gärtner large deviations to a non-Brownian noise setting.
- ItemTransport and continuity equations with (very) rough noise(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Bellinger, Carlo; Djurdjevac, Ana; Friz, Peter; Tapia, NikolasExistence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established.